## 6.7 Measuring strength of trend and seasonality

A time series decomposition can be used to measure the strength of trend and seasonality in a time series (Wang, Smith, & Hyndman, 2006). Recall that the decomposition is written as \[ y_t = T_t + S_{t} + R_t, \] where \(T_t\) is the smoothed trend component, \(S_{t}\) is the seasonal component and \(R_t\) is a remainder component. For strongly trended data, the seasonally adjusted data should have much more variation than the remainder component. Therefore Var\((R_t)\)/Var\((T_t+R_t)\) should be relatively small. But for data with little or no trend, the two variances should be approximately the same. So we define the strength of trend as: \[ F_T = \max\left(0, 1 - \frac{\text{Var}(R_t)}{\text{Var}(T_t+R_t)}\right). \] This will give a measure of the strength of the trend between 0 and 1. Because the variance of the remainder might occasionally be even larger than the variance of the seasonally adjusted data, we set the minimal possible value of \(F_T\) equal to zero.

The strength of seasonality is defined similarly, but with respect to the detrended data rather than the seasonally adjusted data: \[ F_S = \max\left(0, 1 - \frac{\text{Var}(R_t)}{\text{Var}(S_{t}+R_t)}\right). \] A series with seasonal strength \(F_S\) close to 0 exhibits almost no seasonality, while a series with strong seasonality will have \(F_S\) close to 1 because Var\((R_t)\) will be much smaller than Var\((S_{t}+R_t)\).

These measures can be useful, for example, when there you have a large collection of time series, and you need to find the series with the most trend or the most seasonality.

### Bibliography

Wang, X., Smith, K. A., & Hyndman, R. J. (2006). Characteristic-based clustering for time series data. *Data Mining and Knowledge Discovery*, *13*(3), 335–364. https://robjhyndman.com/publications/ts-clustering/